What Determines the Direction of Technological Progress?* Defu Li1

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What Determines the Direction of Technological Progress?  Defu Li1 School of Economics and Management, Tongji University Benjamin Bental2 Department of Economics, University of Haifa Abstract What determines the direction of technological progress is one of the central questions that economics needs to answer. The current paper tries to answer this question by introducing a small but fundamental generalization of Acemolgu (2002). The extended model argues that although changing relative factor prices (as suggested by Hicks 1932) and the relative market size (as argued by Acemoglu 2002) indeed affect the direction of technological progress in the short run, in the long run that direction depends only on the relative supply elasticities of primary factors with respect to their prices. Moreover, it is biased towards enhancing the effectiveness of the factor with the relatively smaller elasticity. The troubling property of the neoclassical growth model discovered by Uzawa (1961), whereby balanced growth is reconcilable only with purely labor augmenting technological progress, is due solely to an implicit assumption that the capital supply elasticity is infinite.

Key Words: direction of technological progress, steady-state, Uzawa’s theorem, investment elasticities, factor supply elasticities.

JEL: O33;O41; E13;E25

 We are grateful to Oded Galor, Daron Acemoglu and Ryo Horii for helpful comments and suggestions on previous versions of the paper. Li gratefully acknowledges support from the National Natural Science Foundation of China (NSFC:71773083), the National Social Science Foundation of China (NSSFC: 10CJL012). The authors take sole responsibility for their views. 1 School of Economics and Management, Tongji University, 1500 Siping Road, Shanghai, P.R. China, [email protected]. 2 Department of Economics, University of Haifa, 199 Aba Khoushy Ave., Mount Carmel, Haifa, Israel, [email protected]. 1

1. Introduction

Technological change can equally increase the productivity of capital and labor, or it can be biased towards a specific factor. For example, according to Kaldor (1961), the stylized characteristics of economic growth in developed countries indicate that while per-capita output and physical capital have grown over time, the capital/output ratio and the income shares of labor and capital have remained basically constant since the industrial revolution.3 These facts have been interpreted as indicating that technological progress has been purely labor-augmenting. In contrast, Ashraf and Galor (2011) show that during the preindustrial era, technological progress has generated population growth and higher density, but not higher per-capita income, which may imply that during that period technological progress was in general not characterized by labor augmentation. Why is it that technological progress had hardly increased labor productivity during the preindustrial era but was focused on labor improvement afterwards? The neoclassical model obtains steady-state growth paths which are consistent with Kaldor’s stylized facts (Solow,1956; Cass,1965; Koopmans,1965). It does so by assuming that technological progress is purely labor-augmenting, but it cannot answer why technological progress must be purely labor-augmenting to exhibit that property.4 On the other hand, when technological progress is assumed to be purely land-augmenting, the Malthusian model’s balanced growth path is consistent with the historical facts confirmed by Ashraf and Galor (2011), but it also cannot explain why technological progress must not include labor-augmentation along a steady-state growth path.5 Moreover, neither of these models can answer the very same questions the other model answers. Acemoglu (2002, 2003) extends the technology of the Romer (1990) model

3These stylized characteristics are further supported by Jones (2015) using the latest available data. 4This problem has troubled growth economists for over half a century ever since the publication of Uzawa’s (1961) famous theorem (Jones and Scrimgeour, 2008; Acemoglu, 2009, pp.59). 5Li and Huang (2016) prove that there is a variant of Uzawa's steady-state theorem in a Malthusian setting. That is, if the model exhibits steady-state growth path, technical change must be purely land-augmenting and cannot include labor augmentation. 2

from one to two dimensions, thereby establishing a framework within which the determinants of the direction of technological progress can be analyzed. However, the model’s extreme restrictions on factor accumulation processes limit its ability to expose what we find to be the key determinants of the direction of technological progress. For the same reason it cannot reconcile the aforementioned discrepant characterizations of pre- and post-industrial revolution technological progress. By removing Acemoglu’s restrictions on the specification of the factor accumulation processes, the current paper not only identifies the determinants of the long-run direction of technological progress, but also provides simple and reasonable answers to the above questions. Specifically, we accept all of Acemoglu’s (2002, 2003) assumptions but allow investment elasticities in the two primary factor accumulation processes to range between 0 and 1, rather than being constrained at either boundary as is commonly done in the literature. This seemingly minor variation of the model is fundamental for analyzing the direction of technological progress. It demonstrates that it is just these elasticities that determine that direction in the long-run, while changes of relative factor prices (as suggested by Hicks 1932) and the relative abundance of these factors (as argued by Acemoglu 2002) impact only the short-run direction of technological progress. 6 Moreover, we show that technological progress is biased towards improving the exploitation of the factor with the relatively smaller elasticity. The intuition behind this result is the following. In the short run, a higher factor price encourages not only invention to economize that factor’s use, but also its accumulation. If the supply elasticity of the factor is very large, it may not be optimal to invest any resources in inventions that economize its use. Furthermore, to offset that factor’s abundance, balanced growth requires an increased investment in technologies that augment the efficiency of the factor with the smaller supply

6 The investment elasticities of the two primary factors in Acemoglu (2002) are set to zero. As a result, only Hick-neutrality is compatible with stationary equilibrium growth path; In Acemoglu (2003) and the neoclassical model the investment elasticity of capital is 1 and in the Malthusian model the investment elasticity of labor is 1. Therefore, in the steady state path, technological progress must be purely labor-augmenting for the former and purely land-augmenting for the latter. Removing these restrictions admits any combination of labor and capital augmentation along a steady-state growth path. 3

elasticity, leading to the extreme configurations of technological progress discussed above. To fix ideas, consider the case of oil. During a long period oil was abundant, and hardly any effort was put into economizing its use, as evidenced by the MPG figures of U.S. produced cars before the 1973 oil crisis. That crisis has caused a sharp increase in oil prices, inducing investment in energy-saving technologies (e.g., increasing MPG). However, the same price increase also induced search for new oil sources, such as shale oil. These new sources have again increased the supply of oil, contributing to sharp price decreases. Consequently, incentives to further invest in energy-saving technologies have decreased.7 With this intuition in mind, the paper suggests the following answers to the aforementioned questions. In the pre-industrial era technological progress did not increase labor productivity because labor supply was very elastic (as described by Malthus 1798). Approximately concurrent with the industrial revolution, the demographic transition reduced the supply elasticity of labor. Moreover, the industrial revolution has replaced land by reproducible physical capital. As the supply elasticity of capital increased, there were no incentives to economize on its use and improve its productivity. Consequently, technological progress was biased towards improving human capital, thereby increasing labor productivity. The ideas in this paper are closely related to previous literature dealing with the direction of technological progress. Over eighty years ago, Hicks (1932) wrote: “A change in the relative prices of the factors of production is itself a spur to invention, and to invention of a particular kind-directed to economizing the use of a factor which has become relatively expensive” (pp. 124-125). Hicks’ ideas were criticized by Salter (1960), basically arguing that since factors are paid the values of their marginal product, cost considerations alone cannot explain the direction of technological progress.

7 According to the PEW Environment Group, the model-year 1975 cars drove about 14 miles per gallon. This figure has doubled by 1985, and stayed roughly stagnant for the next two decades, rising to about 33 by 2005 (see http:// www. pewtrusts.org /~/ media /assets /2011 /04 /history- of -fuel -economy-clean-energy-factsheet.pdf). constructed 4

One important reason for the renewed interest in the direction of technological progress in 1960s was the publication of the aforementioned Uzawa (1961) steady-state theorem. Uzawa proved that if a neoclassical growth model exhibits steady-state growth, then either the production function is Cobb-Douglas or technological progress must be purely labor-augmenting. This result places very uncomfortable restrictions on neoclassical growth theory, as there are no compelling reasons why any of them should hold empirically. Attempting to resolve this issue, Samuelson (1965) and Drandakis and Phelps (1966) constructed growth models whose balanced-growth path is identical to that obtained under pure labor-augmentation. These models were based on Kennedy’s (1964) concept of the “innovations possibilities frontier” which captures the trade-off between different types of innovations.8 Nordhaus (1973) commented that the neoclassical growth models were “saved” by the introduction of the theory of induced innovation, while criticizing its lack of micro-foundations. Because of this shortcoming there was little research on the direction of technological progress for almost thirty years.9 Only the work of Acemoglu (1998, 2002, 2003, 2007, and 2009) which studied the issue using the framework of endogenous technological change (as developed by Romer 1990, and Aghion and Howitt 1992), has renewed interest in this question. Many other authors have also noted the problem raised by the Uzawa theorem (Aghion and Howitt, 1998, pp16; Funk, 2002; Jones, 2005; Jones and Scrimgeour, 2008; Irmen and Tabakovic, 2017; Irmen, 2017a, 2017b). However, all of these papers make restrictive assumptions on investment elasticities similar to Acemoglu’s. Schlicht (2006) provides a very simple proof of the Uzawa theorem, from which it becomes clear that the linear relationship between capital accumulation and investment is the key to the result. The introduction of adjustment costs breaks that linear relationship and allows technological progress to include both labor- and

8 Jones and Scrimgeour (2008) pointed out that these papers just seem to provide an explanation based on the innovation possibilities frontier but do not resolve the problem. 9 For further discussion, see Acemoglu (2002). 5

capital-augmentation along a steady-state growth path (Sato and Ramachandran, 2000; Irmen, 2013). A different approach allowing both types of technological progress has been recently suggested by Grossman et al. (2017).10 However, the main purpose of these papers is to point out the prerequisites for the existence of capital augmenting technological progress, rather than the determination of the factors affecting its direction. The rest of the paper is organized as follows. The second section describes the economic environment of the benchmark model. Building on Acemoglu (2002, 2003), it analyses the behavior of households and firms and characterizes the steady-state equilibrium path; The third section focuses on the determinants of the direction of technological progress and briefly compares the current results to those of the existing literature; The fourth section discusses some applications of the results and some alternative specifications; The fifth section concludes.

2. Benchmark model

The economic environment is an extension of Acemoglu (2002, 2003). The economy consists of two kinds of material factors, and three sectors of production; a final goods sector, an intermediate goods sector and a research and development (R&D) sector. The preference structure, production functions and the innovation possibilities frontier are identical to Acemoglu’s. However, the current analysis differs from that of Acemoglu’s in the factor accumulation functions.

2.1 The economy The following three subsections reproduce Acemoglu (2003) and provide the specification of the underlying structure. 2.1.1 The representative household The representative household owns two kinds of material factors, denote by K

10 One of the features distinguishing the Grossman et al. (2017) paper from the literature surveyed above is its inclusion of embodied technological change in the definition of capital-augmenting technological progress. 6

and L. To facilitate the discussion, we refer to these factors as “capital” and “labor”.11 In addition, the household is endowed by S “scientists” whose role is explained below. The household’s goal is to maximize the discounted flow of utility, given by:

where is consumption at time t, ρ>0 is the discount rate, and θ>0 is a utility curvature coefficient of the household. The household’s periodic budget constraint is given by:

where the LHS stands for expenditures consisting of consumption and investments and into capital and labor, and the RHS is income, obtained from renting out labor at the rate w, capital at the rate r and scientists at the rate .

2.1.2 Production The final goods sector is competitive, using the production function

where Y is output and YL and YK are the two inputs, with the factor-elasticity of substitution given by ε. The factors of production are also produced competitively by constant elasticity of substitution (CES) production functions using a continuum of intermediate inputs, and :

where the elasticity of substitution is given by – and N and M represent the measure of different types of the respective intermediate inputs. For the ease of discussion, we associate the and inputs with respective “labor” or “capital” intensive production technologies, and accordingly interpret an increase in N or in M

11 The “K” should not be think only as physical capital, it also can be interpreted as Land or skilled labor, and the “L” can represent labor, Human capital, or unskilled labor, and so on. 7

as labor- or capital-augmenting technological change. Intermediate inputs are supplied by monopolists who hold the indefinite right to use the relevant patent, and are produced linearly from their respective primary factors:

2.1.3 The innovation possibilities frontier The innovation possibilities frontier functions are given by12

where and represent, respectively, the “number” of scientists who carry out R&D generating “patents” of the labor-and capital-intensive intermediate goods.13 Once an R&D firm invents a new kind of an intermediate input, its patent rights are perfectly enforced and perpetual.

2.1.4 Material factors accumulation While the above follows precisely the Acemoglu (2003) formulation, the factor accumulation processes are different. Specifically, we assume:

，，

，， where the factors K and L depreciates at the rates and , and are the elasticities of investment of factor accumulation. Specifying and to be smaller than 1 represents the idea that converting final output into useable production factors is associated with increasing costs. For example, if K is to be interpreted as physical capital, the transformation may involve adjustment costs that are increasing as investment grows (Irmen, 2013). Similarly, thinking of L as human capital, increasing its size is also likely to be associated with convex education costs.

12 In the extension section below we provide some alternative specifications for which the main results of the paper continue to hold. 13Equation (6) is a simple case of the equation (8) in Acemoglu (2003). 8

Equations (7) generalize the accumulation function of existing growth models. Though the extension is seemly minor, through it the current model becomes a general framework that nests many famous models as special cases. For example, if

and , then in the long run K and L are fixed by , and

, which is equivalent to the assumption of Acemoglu (2002). Setting

and yields the usual cases of the neoclassical growth model (Romer,

1990; Acemoglu, 2003). The case turns out to yield a Malthusian environment. To some extent, the importance of the generalization is surprising. In the later of this paper, it is proven that the two parameters, and , which have been ignored by existing growth models, are the only determinant of the direction of technical progress in the steady-state equilibrium.

2.2 Profit maximization This subsection describes the profit maximization problems of the various actors, replicating Acemoglu (2002, 2003). Letting the final good serve as numeraire, the representative competitive final good producer faces the input prices and and selects the respective and

so as to maximize

subject to the production function (3), yielding the demand functions:

The reperesentative producers of YK and YL maximize their profits by choosing

Z(j) and X(i) given the intermediate input prices and

subject to their respective production functions (4). This generates the demand functions

The intermediate input producers which hold the exclusive right to produce their particular type of input face the prices of the primary inputs and choose, respectively, and to maximize

subject to their technologies (5) and the demand functions (11). From the maximization problem of the intermediate goods producers (12) we obtain:

which imply that all intermediate inputs have the same mark-up over marginal cost. Substituting equations (13) into (11), we find that all capital-intensive and all labor-intensive intermediate goods are produced in equal (respective) quantities.

By the production functions of the intermediate inputs (3), all monopolists have the same respective demand for labor and capital. Finally, the patent holders exctract the monopoly profits from the intermediate goods producers. However, due to the competition for the services of scientists, these profits are paid out as wages, yielding

2.3 Market equilibrium. The material factor market clearing condition implies:

Substituting equations (16) into (4), we obtain the equilibrium quantities of

labor-intensive and capital-intensive inputs:

Finally, using equations (17) and (3), we obtain the amount of the final good:

In order to simplify notation, we follow Acemoglu (2003) by letting and , to obtain:

Therefore, increasing the variety of capital-intensive or labor-intensive intermediate inputs, M and N, implies capital- or labor-augmentation. Let be the ratio of effective capital to effective labor, so that

Accordingly, equation (19) can be rewritten as:

Using equation (21), we transform the market prices of the capital-intensive and labor-intensive inputs (9) into the following forms:

Using equations (16), (17), and (22) in (14), we obtain

Equations (23) indicate that the returns to the primary factors are positively related to the respective “number” of the intermediate inputs. The monopoly profits (12) become, by equations (13), (17) and (23):

Substituting equations (23) into equations (24), we obtain

Equation (25) shows that for a given ratio of the technological levels (M/N), relative invention profits are positively related to the relative factor prices (r/w) and the relative factor supplies (K/L). Accordingly, a change of relative price encourages innovations directed at the scarce factor whose price has increased, as suggested by Hicks (1932). The relative amount of the two factors, (K/L), has two countervailing effects on

. On the one hand, a higher K/L causes an increase in , which in turn leads to a technological change favoring the abundant factor. This is what Acemoglu (2002) named “the market size effect”. On the other hand, a higher K/L decreases

14 and , which is the price effect of a change in K/L. The total effect of a change in K/L is regulated by the elasticity of substitution between the two factors. If , the market size effect dominates the price effect, and increasing K/L will encourage favoring improvements of the abundant factor. Otherwise, when , improvements of the scarce factor will be favored (Acemoglu, 2002). However, holding M/N fixed implies that these effects are only the static or short-run ones. Specifically, when , favoring innovation in the capital-intensive intermediate factor causes M/N to increase. Equation (25) shows that a higher M/N causes to decrease, preventing further inverstment into innovations in the capital-intensive sector. Moreover, equation (25) represents only the demand side of technological change. To get the long effects, it is necessary to consider also factors affecting the supply of innovations and material factors, in particular that of on

K/L and of on , within a dynamic general equilibrium framework. As will be shown below, in such a context, even if there is a short-run “market size effect”, K/L and M/N cannot be both continually increasing in the long-run. Finally, we turn to the market for scientists which determines the supply of innovations. Owing to the free-entry into the R&D sector assumption, the marginal innovation value of scientists should be equal across technologies. Using the

14 It is worth noting that the price effect caused by a change of the relative factors supply K/L is different from the effect of an exogenous change of the relative price r/w when K/L is given. 12

innovation possibilities frontier function (6), this implies

From equation (24) we obtain

Applying equation (21) to (27) yields

Equation (28) shows that market clearing implies that k* is a constant determined solely by the parameters and . Equations (23), (24) and (28), also yield the following factor shares:

Equations (29) show that factor shares are determined solely by the market

15 clearing conditions and depend only on the parameters .

2.4 Consumer behavior Households maximize their objective (1) subject to the budget constraint (2), taking as given the technological change processes (6) and factor accumulation (7). The corresponding Euler conditions are given by equations (30):16

Equations (30) reflect the conditions of the optimal allocation of income among

15 Notice that the market equilibrium is identical to that of Acemoglu (2003). However, that paper does not exploit equation (29) and therefore fails to notice that ε has no impact on factor shares. 16See Appendix A. 13

consumption and the two kinds of investment. The first equation in (30) is the necessary condition for the optimal allocation between physical capital investment and consumption. It is worth noting that when , that equation simplifies to the familiar form . In that environment a constant value of

implies that r must be constant. However, if , when and are constant, the rate r cannot be constant as it must satisfy . Thus steady-state growth does not necessarily imply a constant market rental price of capital. The second equation in (30) is the necessary condition for the optimal allocation between labor investment and consumption. The optimal allocation is achieved when the two equations hold simultaneously. As long as one equation of (30) is not satisfied, the household can obtain a higher level of utility by reallocating its income among consumption and investments. Finally, the transversality condition is given by

2.5 Steady-state equilibrium We summarize the section by stating the conditions implying a steady-state equilibrium for the above environment.

Definition 1: A steady-state equilibrium path (hereafter SSEP) is a dynamic path along which the endogenous variables , are growing at constant rates, household utility and all producer profits are maximized and markets clear at each instant.

Using Definition 1, we obtain the following results:

Proposition 1：For the benchmark economy described above there exists a unique SSEP where equations (32) provide the growth rates of consumption ( ,

output , investments , primary factors and , and the measures of intermediate inputs , and the allocation of scientists and income are given by equations (33).

with , , .

Proof: see appendix B.

From equations (32) and the definition of B and A, the rates of capital- and labor-augmenting technological progress are given by (34):

Corollary 1: If and , then an SSEP includes both types of technological progress.

In this framework, the coexistence of both types of technological progress along an SSEP is due to the introduction of diminishing returns in the factor accumulation processes. Specifically, when < 1, there is a gap between capital and output growth rates (see the second equation of (32)). That gap is closed by capital-augmenting technological progress (the first equation of (34), see also Irmen

2013). Similarly, when , labor is accumulated at a rate that falls short of output growth, and labor-augmentation makes up for the difference. Notice also that in the current framework the presence of both types of technological progress does not contradict the fact that factor shares remain constant (see equations (29) above).17 The stationary equilibrium solution enables us to infer what determines the direction of technological progress along an SSEP, which is the topic of the next section.

3. The determinants of the direction of technological progress

Identifying what determines the direction of technological progress is the main objective of this paper. Before stating the results we give a clear definition of that direction.

Definition 2: The direction of technological progress, DT, is the ratio between the rates of capital-and labor-augmenting factors, i.e. .

When and then , and technological progress is purely labor-augmenting (i.e. Harrod-neutral); when and then , and technological progress is purely capital-augmenting (i.e.

17This stands in contrast to Samuelson (1965), Drandakis and Phelps (1966), and Acemoglu (2003）who argued that labor-augmenting technological progress is one of the main assumptions needed to explain the stability of factor shares. 16

Solow-neutral); when then , and technological progress is Hicks-neutral. Figure 1 shows different directions of technological progress:

E B  C B/ B

A D 4

O 5° A B Figure 1: Direction of technological progress

Clearly, the axes represent Harrod-neutral (horizontal) and Solow-neutral (vertical) technological changes. The diagonal represents the location of

Hicks-neutral technological changes. The ray indicates technological progress which tends to be more labor augmenting, while is more capital augmenting.

3.1 Main results Using definition 2, we can state the main results of this paper.

Proposition 2: Along an SSEP equations (34) immediately imply：

Equation (35) shows that the direction of technological progress is determined by the investment elasticities of the primary factor accumulation, namely and which have been ignored by the existing growth models surveyed in the introduction, even though they have been designed to analyze the direction of technological progress. This is also why growth theorists have been puzzled by the Uzawa theorem for so long. From the equation (34) we know if ( , there will be no capital-augmentation (labor-augmentation) in steady-state growth. The same equation

implies that it is due to specifying and that the steady-state technological progress of the Acemoglu (2002) model must be Hicks neutral. It also shows that the parameters of the production function and the innovation possibilities frontier, such as the elasticity of substitution between labor and capital, , do not affect the direction of technical progress. In order to provide an economic intuition for equation (35), we define next the primary factors’ supply elasticities and then discuss the relationship between these elasticities and the direction of technological progress.

Definition 3: The supply elasticity of any primary factor X with respect to its price p is given by

With this definition in mind, we obtain the following relationships:

Corollary 2: Along an SSEP, the supply elasticities of capital and labor are given by:

The result follows immediately from equations (32) alongside the time derivatives of (23) (using equation (34) and remembering that k* is constant). Along an SSEP, equations (37) show that the factor supply elasticities are determined by the investment elasticities in the respective accumulation processes. This is because and regulate the degree to which returns to investment in factor accumulation are diminishing. Specifically, the higher or are, the higher are the returns to the respective investment. As a result, the quantitative response to a price change will increase, i.e. the supply elasticity will be higher. Using equations (37) in (36) directly obtains:

The interpretation of equation (38) is summarized as Proposition 3 which is the key result of the paper.

Proposition 3: Along an SSEP the direction of technological progress is determined solely by the relative primary factor supply elasticities and is biased towards the one with the relatively smaller elasticity.

Proposition 3 and equation (38) show that the direction of technological progress is not biased towards the relatively more or less abundant factor, but rather towards the harder to accumulate one. In other words, if one factor is relatively harder to accumulate, balanced growth requires that it must be augmented by technological change.

3.2 Comparison with Hicks (1932) and Acemoglu (2002). Hicks (1932) argued that a change in the relative prices of the factors of production spurs invention and Acemoglu (2002) suggested that the relative market sizes is another factor that shapes the direction of technological progress. However, equations (35) and (38) show that when the economy is on an SSEP neither appears as a determinant of that direction, as stated by the following:

Proposition 4: Along an SSEP the direction of technological progress remains unchanged despite the continually changing relative factor prices and relative factor supplies. Proof: Using equations (32) and applying equation (37), given initial values

which are on an SSEP, the time evolutions of the relative primary factor supply (K/L), the relative price (r/w) and the relative technology level (B/A) are given by:

where , , are the respective initial values of these variables. Equations (39) show that (K/L), (r/w) and (B/A) are evolving along the SSEP and that their growth (or decline) rates are impacted by the relative size of the primary factor price elasticities. Since the latter stay constant by equations (37), so does DT by equation (38). ■ It is important to notice the distinction between growth and level effects. The direction of technological progress refers to the relative change in the capital- and labor-augmenting processes. However, even if this relative change is constant, the relative levels of the two technologies are changing (unless DT=1). Consider for example the case . In that case, the second equation of (39) reveals that the price of labor, w, increases faster than that of capital, r. According to the Hicksian hypothesis this should induce more labor-augmenting technological progress thereby reducing B/A. This logic is fully consistent with the current model, as can be seen from the last equation of (39). However, in this case, the first equation of (38) shows that K/L will keep increasing along an SSEP, while the relative technology B/A will keep decreasing. This is because k*=BK/AL stays constant, so that a rising K/L is consistent only with a declining B/A. Accordingly, in the current framework Acemoglu’s (2002) case, where an increase in K/L raises B/A, may happen only along a transition path, that is, only when the economy moves from one SSEP to another. Specifically, in the benchmark

model k* may be increasing from to due to some change in the exogenous 18 parameters, such as , , or . Once arriving at a new steady state, the

18Acemoglu (2002, 2009) assumes that K and L are given and provides the determinants of the relative technology levels (B/A). In that environment, k* may be changing due to exogenous changes in K or L. As argued 20

direction of technological progress will again be determined by the equation (38).19

The above distinction also highlights the difference between a static and a dynamic concept of factor scarcity. In the static sense, a factor is relatively scarce if its quantity is smaller than that of the other. Acemoglu’s (2002) aforementioned “market size effect” which implies that technological progress will favor the relatively more abundant factor, (see discussion below equation (25)), relates to that static scarcity sense. In the dynamic sense, a factor is relatively scarce if it is harder to accumulate, and by equation (37) has a smaller supply elasticity. In this sense it is the relatively scarce factor that enjoys the faster technological augmentation in the long-run.

4. Discussion and Extension

This section first provides a possible interpretation of the model’s results, and then turns to some possible extensions. 4.1 The role of the supply elasticities

Equation (37) implies that when , . Furthermore, from equation (34) we get , i.e. there is no labor augmentation. In addition, equations (32) imply that Y and L grow at the same rate. It is in this sense that this SSEP is Malthusian. Labor supply is perfectly elastic, and while output may be growing due to capital (or land) augmenting technological change, labor grows just as fast, leaving no room for per-capita increases in income and consumption. In fact, many have argued that this feature characterizes, to a large extent, the growth path prior to the industrial revolution (see, e.g, Ashraf and Galor 2011).

In a similar vein, let . Clearly equations (7) imply that the capital

above, fixing K and L amounts to setting .However, we have seen that with , and B/A will be continually changing. 19In fact, in all cases considered by Acemoglu (2002) technological progress is Hicks-neutral (that is, DT=1) and not affected by changes in K/L along the steady-state path. 21

accumulation process is linear in investment, which is the standard assumption in neoclassical growth models. From equations (32) we obtain that the capital/output ratio is constant, and equations (34) imply that there is no capital augmentation. These features are in line with the Kaldor (1961) stylized facts. Equation (37) implies in

20 addition that . To summarize, the current model is consistent with the “Kaldor facts” as long as and is finite. A Malthusian path in which per-capita income remains constant is obtained if . Additional conditions often found in the literature stating that a Malthusian path requires K (interpreted as “land”) to be constant and technological progress to be nil do not apply in the current framework. Moreover, as shown above, the model finds that both types of technological progress may coexist with a bias towards labor (as long as .This finding is consistent with empirical studies. Sato’s (1970) analysis of 1909-1960 US national income data pointed out that technological progress increased both capital and labor productivity, with labor efficiency increasing faster than capital’s. More recently, Doraszelski and Jaumandreu (2015) used Spanish industrial panel data and found that at the firm level technological change was labor-biased.21 4.2 Possible extensions The benchmark model is based on several assumptions concerning the underlying technologies. In particular, technological change is assumed to take the form of invention of new goods; R&D requires only the input of “scientists”, i.e. that sector does not compete for investment goods; and the productivity in the R&D sector depends only on the number of existing goods in the same sector. It turns out that the key results (equations (35) and (38)) may be obtained under somewhat different specifications of the innovation technologies, albeit subject to some knife-edge

20Acemoglu (2003) seems to suggest that technological progress must be labor-augmenting because “capital, K, can be accumulated, while labor, L, cannot.” The current model shows that technological progress is purely labor-augmenting not because labor cannot be accumulated but because the supply elasticity of capital is infinite. 21 Sato (1970) reports yearly labor augmentation of about 2%, and capital augmentation of roughly 1.3%. Doraszelski and Jaumandreu (2015) decompose technological change into Harrod- and Hick-neutral components, finding that both have increased by an annual rate averaging 2% . In our terms this would translate into yearly rates of 4% labor- and 2% capital-augmentation. 22

conditions.

4.2.1 Knowledge spillover The first extension takes the knowledge spillover model used in Acemoglu (2002,2009). According to that model, productivity in any of the R&D sectors depends on the number of existing varieties in both sectors. Specifically, the innovation possibilities frontier is defined by

，＋＝

The above benchmark model is obtained when , which Acemoglu called “extreme state dependence”.

Proposition 5：If the innovation possibilities frontier (6) is replaced by equations (40) with , while keeping the remaining assumptions of the benchmark model, an SSEP exists only under the knife-edge condition of .

Proof: see appendix C.

Corollary 3: Under the conditions of Proposition 5, the direction of technological progress is determined by equation (35) and (38).

The intuition of the result follows directly from equations (40). To keep the growth rates of M and N constant, it must be the case that M/N is constant, so that they both grow at the same rate. This implies that B and A also grow at the same rate. As a result, K and L grow at the same rate, which requires .Clearly, by equation

(37) we obtain and technological progress must be Hicks-neutral.

4.2.2 Lab equipment model The lab equipment model was suggested by Rivera-Natiz and Romer (1991), and

is used in Acemoglu (2002, 2003, 2009). In that model, the main input into the R&D sectors is final output. As a result, the accumulation processes and the R&D sectors compete for resources. To investigate the impact of this competition the current subsection generalizes the lab equipment model of Acemoglu (2002, 2003, 2009) and assumes the following innovation functions:

，，

where and are investments needed to develop new varieties M and N of the respective intermediate inputs, and and are respectively deprecation rates of blueprints of new varieties of capital- and labor-intensive intermediate inputs.22In this setting, the representative household’s income can be used for either consumption or the corresponding four types of investments. In addition, the households are the direct owners of the patents and accordingly obtain the monopoly profits of the intermediate input producers. In this case too, for an SSEP path to exist, some knife-edge conditions must prevail among the parameters as summarize by:

Proposition 6: Subject to the proper modifications of the benchmark model, if the innovation possibility frontier takes the form of equations (41), an SSEP exists only under the following knife-edge conditions:23

(42)

Proof: see appendix D.

Corollary 4: Subject to conditions (42), the direction of technological progress

22 When and , in the long run M and N will be fixed at and , respectively. When , equations (40) are as same as the equation (19) (without depreciation) in Acemoglu (2002) or equation (34) in Acemoglu (2003). However, in the Acemoglu (2002, 2003) cases, there exists no SSEP unless which is Acemoglu (2002)’s assumption, in contrast to and as assumed by Acemoglu (2003). 23 Knife-edge conditions are often found in the growth literature. See, e.g., Jones (1995); Christiaans (2004); Growiec (2010); Grossman et al. (2016). 24

is determined by equation (35) and (38).24

4.3. Amending the input production function

The production functions of the inputs YL and YK of equation (4) may be replaced by identical constant elasticity of substitution (CES) production functions with corresponding intermediate inputs, X(i) and Z(i):

Once invented, all kinds of intermediate inputs can be produced at a fixed marginal cost in terms of the final output, as in Acemoglu (2009). It can be shown that under the specification (43), the associated fixed marginal production cost of the intermediate inputs and the remaining assumptions of the benchmark model, the direction of technological progress is still determined by equations (35) and (38).25

6. Conclusions

What determines the direction of technological progress? This is one of the central issues of macroeconomics, development economics, labor economics, and international trade. By relaxing the crucial restrictions on the investment elasticities (or supply elasticities) of factor accumulation, the Acemoglu (2002, 2003) framework is extended and a clear answer to the question is obtained. The minor but fundamental relaxation results in the following substantive contributions to the literature: First, it identifies the determinants of the direction of technological progress which are largely ignored by existing growth models. It was shown that despite their short-run impact pointed out by Acemoglu (2002), relative factor prices and the market size effects are not playing any role as long-run determinants of the direction of technological

24 Proof is available upon request. 25 Proof is available upon request. 25

progress. Instead, along a stationary equilibrium path, it is the relative size of the supply elasticities of material factors with respect to their prices which determines that direction, biasing it towards the factor with the relatively smaller elasticity. Second, it suggests a reasonable answer to the question why technological progress was purely land-augmenting in the preindustrial era and purely labor-augmenting after the industrial revolution. The paper argues that this is due to the very high labor supply elasticity in a Malthusian world on the one hand, and very high renewable physical capital supply elasticity but much lower labor supply elasticity after the industrial revolution. Third, it presents a simple resolution to the dilemma of Uzawa’s theorem since the steady state path is compatible with any type of technical change (including any labor- and capital-augmenting), not just for a Cobb-Douglas production function. Fourth, it nests several famous growth models as special cases.

For example, amounts to Acemoglu (2002), and

amounts to Acemoglu (2003) and the neoclassical growth model, and may be interpreted as a Malthusian environment with “K” standing for "land". The paper is also relevant for the discussion concerning the possible causes of the global decline in labor shares and increased income inequality (e.g., Karabarbounis and Neiman, 2013; Piketty, 2014). Some authors have argued that it is the bias of technological progress towards capital-augmentation which causes these phenomena. However, the results presented above cast doubt about this conclusion as they imply that there is no necessary connection between capital-augmentation and declining labor shares even in transition path. There is a number of questions we have not addressed. We provided no micro-foundation for the diminishing investment elasticities in the factor accumulation functions. A theory along these lines is also necessary to explain how the industrial revolution caused technological progress to change its direction from not increasing labor productivity to only increasing that productivity. Another key phenomenon not addressed by the current framework is the continued decline of the relative price of investment goods (Karabarbounis and 26

Neiman, 2013; Grossman et al., 2017). This trend indicates that embodied technological progress has been playing a role for quite some time. The introduction of this option is likely to affect the resource allocation of profit-maximizing R&D firms between factor-augmentation and embodied technologies. All of these are important problems for fruitful future research.

Appendix A： Deriving the Euler Equations (30) Let the Hamilton associated with the optimization problem be:

The first-order conditions are:

Taking log-derivatives of both sides of (A2) over time, we obtain

The motion equations of λ are:

Based on (A2) and (A4), we obtain

Using (A5) in (A3), we obtain the Euler equations (30).

Appendix B: Proof of Proposition 1. We first conjecture there is a SSEP then verify it indeed exists by solving for it. First, we prove that there is a SSEP given by equations (30). From the budget constraint (2) and Definition 1, we obtain

Then, according to the primary factor accumulation functions and (7), the along an SSEP the following must hold:

From equation (21) we can obtain

Since k is constant on the SSEP, from (B3) we get

Equations (B1), (B2), (B4) together with the innovation possibilities frontier (6), yield:

From (B5) and ＋＝ , we obtain the allocation of scientists between two kinds of intermediate R&D given by (B6).

Combining (B1) ,(B5) and (B6), we get the growth rates as given by (B7).

Substituting (B6) into the innovation possibilities frontier (6), and (B7) into (B2) we obtain

(B7) and (B8) confirm that the benchmark model indeed has a SSEP. While (B6) shows that there exist also an allocation of scientists which supports the SSEP, it still needs to be verified that there exists an appropriate allocation of income as given in (33). Using equations (23), the Euler equations (30) can be written as:

Define , . Substituting (B1), , , the definitions (20) and (21), and rewriting (7) we get:

＋

Insert (B1), (B2) into (B10) to obtain

＋

Rearranging (B11) yields:

Using in from (28), equation (B12) is rewritten as:

Inserting (B7) into (B13), we obtain

Define so that:

Inserting (B14) in (B15) obtain that along a in SSEP, given by

Equations (B6), (B14) and (B16) given the allocation of scientists and income to reach the SSEP given by (32). Finally, notice that the solution process implies that there exists only one allocation of scientists and income that is consistent with a SSEP. ■

Appendix C: Proof of Proposition 5. From equation (40) we can obtain

To keep the economy on a SSEP, . This implies

Furthermore,

Using equation (C3) in equation (B4) we get

Applying (B2) to (C4), we obtain that in this case there exist a SSEP under the knife-edge condition

■ Appendix D: Proof of Proposition 6

If , then there may be , where ; if

then there may be , ; if then there may be , . At these cases, equation (D1) cannot be consistent with the modified budget constraint and the definition of SSEP. So the proof of the necessary condition for the existence of a SSEP will include three steps. First, we prove it exists when , and if ; Second we prove it exists when , where ; third, we prove it exists when

, and .

First, if , and if then from the modified budget constraint and the definition of a SSEP, we obtain

Then, according to the factor accumulation processes (7) and the innovation possibilities frontier (41), the following equations must hold along a SSEP:

Using the intensive form of the production function (21), we obtain

In a SSEP, due to the fact that k is constant, we have:

Substituting (D1), (D2) and (D3) into (D5), if we obtain the

necessary condition for the existence of a SSEP of equation (42)

Second, if , then , and in the long run . Then from (D5) we obtain

From (D3) we obtain

From (D7) we get . And when , from the modified budget constraint we have .Using this in (D7) we get

From (D8) we also have .

Similarly, we can prove that if then must be equal to , and if , where , then it must be that , where . In all these cases, equation (42) must hold if a SSEP is to exist.

Third, if then , and the modified budget constraint implies:

Using (D9) in (D5) we get . From the innovation possibilities frontier (41) we know that only two possible cases can attain . One is

.Then , the other case is and .

However, from the latter case we can get

As a result, will also be a constant. However, if , then cannot be a constant. So if then is the only possible way to get in the SSEP. As a result, we also obtain .

Similarly, we can prove that if then it must be that .If

, where , then it must be that , where where . In all these cases equation (42) must hold.

To summarize, from the above three steps, we obtain when and

, equations (42) must hold if a SSEP is to exist. ■

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